In this example, we show that any square matrix with complex entries can uniquely be decomposed into the sum of one Hermitian matrix and one skew-Hermitian matrix. A fancy way to say this is that complex square matrices is the direct sum of Hermitian and skew-Hermitian matrices.

Let us denote the vector space (over $\mathbb{C}$ ) of complex square $n\times n$ matrices by $\matC$ . Further, we denote by $\matCp$ respectively $\matCm$ the vector subspaces of Hermitian and skew-Hermitian matrices. We claim that \begin{eqnarray} \label{eqp} \matC &=& \matCp \oplus \matCm. \end{eqnarray}Since $\matCp$ and $\matCm$ are vector subspaces of $\matC$ , it is clear that $\matCp +\matCm$ is a vector subspace of $\matC$ . Conversely, suppose $A\in \matC$ . We can then define \begin{eqnarray*} A_+ &=& \frac{1}{2}\big( A + A^\ast \big), \\ A_- &=& \frac{1}{2}\big( A - A^\ast \big). \end{eqnarray*}Here , and $\ccj{A}$ is the complex conjugate of $A$ , and is the transpose of $A$ . It follows that $A_+$ is Hermitian and $A_-$ is anti-Hermitian. Since $A=A_+ + A_-$ , any element in $\matC$ can be written as the sum of one element in $\matCp$ and one element in $\matCm$ . Let us check that this decomposition is unique. If $A\in \matCp\cap \matCm$ , then $A=A^\ast=-A$ , so $A=0$ . We have established equation .

Special cases

• In the special case of $1\times 1$ matrices, we obtain the decomposition of a complex number into its real and imaginary components.
• In the special case of real matrices, we obtain the decomposition of a $n\times n$ matrix into a symmetric matrix and anti-symmetric matrix.